New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions.

*(English)*Zbl 1205.65216This paper is concerned with the approximate solution of second order differential equations \( u''(t) + \sigma u'(t) + f(t, u(t)) = 0,\) \(t \in [0,1],\) where \( \sigma \) is a non zero constant and \( f: [0,1] \times {\mathbb R} \to {\mathbb R} \) is a sufficiently smooth function, supplemented with linear integral boundary conditions of type: \( u(0) - \mu_1 u'(0) = \int_0^1 h_1(s) u(s) ds,\) \( u(1) + \mu_2 u'(1) = \int_0^1 h_2(s) u(s) ds, \) with positive constants \( \mu_j\) and given smooth functions \( h_j(t)\).

The proposed approach starts establishing an homotopy defined by family of differential equations \( H(u,p) \equiv u'' + \sigma u' + p f(t,u) = 0\), with the parameter \( p \in [0,1]\) so that for \( p=0\) gives a linear equation such that with the boundary conditions has a unique solution \( u = u_0(t)\) easily computed and for the parameter value \( p=1\) is the desired solution of the non linear problem. Now by using \( p\) as a small parameter the solution of \( H(u,p)=0\) can be written as an asymptotic series \( u=u_0+ p u_1 + \dots \) where the successive \( u_j\) can be computed recursively as a solution linear problems and then the solution for \( p=1\) is approximated by the \((m+1)\)-sum \( u = \sum_{j=0}^m u_j\). For solving each linear boundary value problem of \( u_j\) the authors propose a reproducing kernel Hilbert space method. Two numerical experiments are presented to show the behaviour of the method depending on the terms \(m\) of the series and the number of grid points in the interval \([0,1].\)

The proposed approach starts establishing an homotopy defined by family of differential equations \( H(u,p) \equiv u'' + \sigma u' + p f(t,u) = 0\), with the parameter \( p \in [0,1]\) so that for \( p=0\) gives a linear equation such that with the boundary conditions has a unique solution \( u = u_0(t)\) easily computed and for the parameter value \( p=1\) is the desired solution of the non linear problem. Now by using \( p\) as a small parameter the solution of \( H(u,p)=0\) can be written as an asymptotic series \( u=u_0+ p u_1 + \dots \) where the successive \( u_j\) can be computed recursively as a solution linear problems and then the solution for \( p=1\) is approximated by the \((m+1)\)-sum \( u = \sum_{j=0}^m u_j\). For solving each linear boundary value problem of \( u_j\) the authors propose a reproducing kernel Hilbert space method. Two numerical experiments are presented to show the behaviour of the method depending on the terms \(m\) of the series and the number of grid points in the interval \([0,1].\)

Reviewer: Manuel Calvo (Zaragoza)

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |

34B30 | Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) |

##### Keywords:

second order ordinary differential equations; integral boundary conditions; perturbation methods; reproducing kernel Hilbert space method (RKHSM); numerical experiments; homotopy perturbation method (HPM)
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\textit{F. Geng} and \textit{M. Cui}, J. Comput. Appl. Math. 233, No. 2, 165--172 (2009; Zbl 1205.65216)

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